It was discovered in the early XX century that the description of matter at low-scales demands to wave-off some of the existing at the time
paradigms governing the motion and very structure of particles in interactions. This led to the development of the theory of relativity on the one hand, and
quantum mechanics on the other. In the latter setting, the state of a physical system is described by a vector, the wave function, belonging to some Hilbert space
and supposed to encapsulate all the physical degrees of freedom of that system.
On the classical level, the time evolution of
particles' momenta and positions is governed by a set of generically non-linear ordinary differential equations which can be written in the form of Hamilton's equations.
In its turn, the time evolution of a wave function is governed by a first order ordinary linear differential equation driven by the Hamiltonian operator.
This operator is obtained through a quantisation procedure: its symbol is given by the classical Hamiltonian of the system
or, said differently, it is obtained from the classical Hamiltonian upon replacing the classical momenta and positions by operators.
While the success of the approach was astonishing relatively to the amount of experiments which could
have been explained, soon after the early development of the theory it became clear that in order to describe physics at even smaller scales
or higher energies, one needs to develop a quantum theory of fields which would bring together the quantum and relativistic features in the setting of uncountably many degrees of freedom.
In loose words, such a theory would be reached by producing operator valued generalised functions, viz. formal kernels of
distributions, depending on the space-time coordinates which would satisfy analogues of non-linear, relativistically invariant, evolution equations arising in classical field theory.
While it was rather straightforward to construct the quantum theory of the free field (and nowadays such a construction is fully rigorous)
the construction of interacting theories, the sole relevant for physics, appeared to be a tremendously hard task, this even on a formal level of rigour.
The various approaches that were developed quickly met serious problems, the most prominent being the divergence of coefficients supposed to describe the formal perturbative expansions of physical observables
around the free theories. Eventually, these problems could have been formally circumvented in certain cases by the use of the so-called renormalisation procedure.
The latter, while being able to produce numbers which were measured with great agreement in collider experiments, eluded for very long any
attempts at making it rigorous. Some progress was eventually achieved for several instances of truly interacting, viz. non-free, quantum field theories
within the so-called constructive quantum field theory approach, see [1]} for a review. While successful in rigorously showing
the existence and certain overall properties of such theories, the approach did not lead yet to rigorous and manageable expressions for the correlation functions,
which are the quantities measured in experiments and thus of prime interest to the theory.
